Let $ \vec\Omega, \vec\Omega' $ be two unit vectors in $ \mathbb R^3 $ such that $ \vec\Omega\cdot\vec\Omega' = t $ and let $ r > 0 $. The multipole expansion for the exponential reads:
$$ \mathrm e^{\mathrm i r t} = 4\pi \sum_{l=0}^\infty \sum_{m=-l}^l \mathrm i^l j_l(r)Y_{lm}(\vec\Omega)Y^*_{lm}(\vec\Omega') \\ = \sum_{l=0}^\infty (2l+1)\mathrm i^l j_l(r) P_l(t). $$
In Mathematica this series is given by:
Sum[(2 l + 1) I^l SphericalBesselJ[l, r] LegendreP[l, t], {l, 0, ∞}]
However, it seems that Mathematica is unable to compute this and similar series, for example:
$$ \sum_{l=0}^\infty (2l+1) j_l^2(r) P_l(t) = \mathrm{sinc}(r\sqrt{2-2t}). $$
Is there any way to fix it?
w[r_, t_] := Sum[(2 l + 1)*SphericalBesselJ[l, r]^2*LegendreP[l, t], {l, 0, 25}] // N; Plot3D[{Sinc[r*Sqrt[1 - t]], w[r, t]}, {r, 0, 2}, {t, 0, 2}]
??? $\endgroup$w[r_, t_] := Sum[(2 l + 1)*SphericalBesselJ[l, r]^2*LegendreP[l, t], {l, 0, 25}] // N; Plot3D[{Sinc[r*Sqrt[2 - 2 t]], w[r, t]}, {r, 0, 2}, {t, 0, 2}]
$\endgroup$